Global Well-posedness of an Inviscid Three-dimensional Pseudo-Hasegawa-Mima Model

TL;DR

This paper introduces a simplified pseudo-Hasegawa-Mima model inspired by plasma turbulence equations, proving its global well-posedness and continuous dependence on initial data, thus advancing mathematical understanding of related plasma models.

Contribution

The paper establishes the first rigorous results on existence and uniqueness for a simplified 3D inviscid plasma turbulence model, using techniques from Euler equations.

Findings

Proved global existence and uniqueness of solutions.

Established continuous dependence on initial data.

First such results for this class of plasma models.

Abstract

The three-dimensional inviscid Hasegawa-Mima model is one of the fundamental models that describe plasma turbulence. The model also appears as a simplified reduced Rayleigh-B\'enard convection model. The mathematical analysis the Hasegawa-Mima equation is challenging due to the absence of any smoothing viscous terms, as well as to the presence of an analogue of the vortex stretching terms. In this paper, we introduce and study a model which is inspired by the inviscid Hasegawa-Mima model, which we call a pseudo-Hasegawa-Mima model. The introduced model is easier to investigate analytically than the original inviscid Hasegawa-Mima model, as it has a nicer mathematical structure. The resemblance between this model and the Euler equations of inviscid incompressible fluids inspired us to adapt the techniques and ideas introduced for the two-dimensional and the three-dimensional Euler equations to prove the global existence and uniqueness of solutions for our model. Moreover, we prove the continuous dependence on initial data of solutions for the pseudo-Hasegawa-Mima model. These are the first results on existence and uniqueness of solutions for a model that is related to the three-dimensional inviscid Hasegawa-Mima equations.